In the circle below segment AB is a diameter. If the length of arc ACB is 6pi , what is the length of the radius of the circle?
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### Geometry Problem Example
**Question:**
In the circle below, segment AB is a diameter. If the length of arc \( \overset{\frown}{ACB} \) is \( 6\pi\), what is the length of the radius of the circle?
**Explanation:**
In the figure, you will see a circle with points A, B, and C on its circumference. The segment AB is indicated as a diameter, passing through the center of the circle and dividing it into two equal parts. The arc \( \overset{\frown}{ACB} \), which is a section of the circle's circumference, is highlighted and its length is given as \( 6\pi \).
### Diagram Details:
- **Points**: A and B are the endpoints of the diameter.
- **Arc**: \( \overset{\frown}{ACB} \) represents the arc from point A to point B passing through another point C on the circle.
- **Diameter**: The straight line segment AB passing through the center of the circle.
### Calculation:
Given that the arc \( \overset{\frown}{ACB} \) is \( 6\pi \), we can use the formula for the circumference of a circle, which is \( C = 2\pi r \), where \( r \) is the radius:
1. Observe that arc \( \overset{\frown}{ACB} \) is half of the circle's circumference, since AB is the diameter and splits the circle into two equal arcs.
2. Therefore, the length of the full circumference \( C \) would be twice the length of the arc \( \overset{\frown}{ACB} \), since it only represents half of the circle.
\[ C = 2 \times 6\pi = 12\pi \]
3. Using the circumference formula \( C = 2\pi r \), substitute the given value:
\[ 12\pi = 2\pi r \]
4. Solve for \( r \):
\[ r = \frac{12\pi}{2\pi} = 6 \]
Therefore, the radius of the circle is 6 units.
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This solution includes step-by-step mathematical reasoning essential in solving problems involving circles' geometry. Understanding these principles is beneficial in various practical and theoretical applications in mathematics.
Two-dimensional figure measured in terms of radius. It is formed by a set of points that are at a constant or fixed distance from a fixed point in the center of the plane. The parts of the circle are circumference, radius, diameter, chord, tangent, secant, arc of a circle, and segment in a circle.
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**Explanation:**
In the figure, you will see a circle with points A, B, and C on its circumference. The segment AB is indicated as a diameter, passing through the center of the circle and dividing it into two equal parts. The arc \( \overset{\frown}{ACB} \), which is a section of the circle's circumference, is highlighted and its length is given as \( 6\pi \).
### Diagram Details:
- **Points**: A and B are the endpoints of the diameter.
- **Arc**: \( \overset{\frown}{ACB} \) represents the arc from point A to point B passing through another point C on the circle.
- **Diameter**: The straight line segment AB passing through the center of the circle.
### Calculation:
Given that the arc \( \overset{\frown}{ACB} \) is \( 6\pi \), we can use the formula for the circumference of a circle, which is \( C = 2\pi r \), where \( r \) is the radius:
1. Observe that arc \( \overset{\frown}{ACB} \) is half of the circle's circumference, since AB is the diameter and splits the circle into two equal arcs.
2. Therefore, the length of the full circumference \( C \) would be twice the length of the arc \( \overset{\frown}{ACB} \), since it only represents half of the circle.
\[ C = 2 \times 6\pi = 12\pi \]
3. Using the circumference formula \( C = 2\pi r \), substitute the given value:
\[ 12\pi = 2\pi r \]
4. Solve for \( r \):
\[ r = \frac{12\pi}{2\pi} = 6 \]
Therefore, the radius of the circle is 6 units.
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This solution includes step-by-step mathematical reasoning essential in solving problems involving circles' geometry. Understanding these principles is beneficial in various practical and theoretical applications in mathematics.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Feb91831e-83e8-4bb6-ba90-3985f8038ff0%2Fa1f091d9-efde-4b26-910e-6ef5dadb5e05%2Fpissu_processed.jpeg&w=3840&q=75)
